Triangle Inequality my balls

Sup /sci/.

well I had this problem twice in 2 days while solving some shit on L^p spaces, I can't figure if |x+y|^p =< |x|^p + |y|^p . Seems quite good to use it like this is true, but it seems rather false when I look at an example : if x and y are positive, |x+y|^p = (x+y)^p = Newtonfag(x,y) >> x^p + y^p...

Problem is I saw this used in a demonstration of Clarkson inequality which is quite simpler than the one I know and use. It uses the fact that |x+y|^p =< |x|^p + |y|^p , |x-y|^p =< |x|^p + |y|^p and conclude. The second one may be right since the minus is there but I can't accept the first.

As I saw this shit in some über book, I might have overlooked something or he writers are just faggots. Some proof/advice/confirmation ?

Tits can't be granted but I may answer with an ASCII tits.

It's not true if you mean the absolute value of x+y to the pth power. But that's not what you actually meant, is it? What you actually meant is that the pth-norm of x+y is less than the pth norm of each individually.

In L^p spaces, this becomes (|x+y|^p)^1/p =< (|x|^p)^1/p + (|y|^p)^1/p. Try working with that and see if you can understand it generally for all p>=1. You might even try the cases of p=1 and p=2 to just make sure that it is true.

Also note, when x is n dimensional, |x|^p becomes |x_1|^p+...+|x_n|^p.